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Integer and rational exponents — ACT practice questions

Integer and rational exponents on the ACT Mathematics test cover integer, negative, and fractional powers and scientific notation. Students must rewrite products of powers, compare exponential quantities, and simplify expressions that hold for all nonzero u and v. Items often ask which listed form is equivalent, which quantity is greatest, or how many liters remain after 4.8 × 10^6 liters drain at 7.5 × 10^4 liters per hour for 8 hours. Answer choices are short numeric or algebraic expressions, and traps include adding exponents that should be multiplied, mishandling negative exponents, and writing scientific notation with the wrong power of ten.

  • Section: Mathematics 45 questions on the paper
  • 41 practice questions in the app

Sample questions

Question 1

Which of the following is equivalent to 18u12v53u4v2\dfrac{18u^{12}v^5}{3u^4v^2} for all nonzero uu and vv?

  1. A6u3v36u^3v^3
  2. B15u8v315u^8v^3
  3. C6u16v76u^{16}v^7
  4. D6u8v36u^8v^3
Show answer

Answer: D — 6u8v36u^8v^3

The coefficient is 18÷3=618\div3=6. Subtracting exponents gives u12−4v5−2=u8v3u^{12-4}v^{5-2}=u^8v^3, so the ratio is 6u8v36u^8v^3.

Question 2

Which of the following is the greatest?

  1. A9−1/29^{-1/2}
  2. B8−1/38^{-1/3}
  3. C16−116^{-1}
  4. D25−1/225^{-1/2}
Show answer

Answer: B — 8−1/38^{-1/3}

The four values are 13\dfrac{1}{3}, 12\dfrac{1}{2}, 116\dfrac{1}{16}, and 15\dfrac{1}{5}, respectively. The greatest is 8−1/3=128^{-1/3}=\dfrac{1}{2}.

Question 3

A reservoir initially contains 4.8×1064.8 \times 10^6 liters of water. Water drains at a constant rate of 7.5×1047.5 \times 10^4 liters per hour for 8 hours. How many liters remain, expressed in scientific notation?

  1. A4.725×1064.725 \times 10^6
  2. B5.4×1065.4 \times 10^6
  3. C4.2×1064.2 \times 10^6
  4. D4.2×1054.2 \times 10^5
Show answer

Answer: C — 4.2×1064.2 \times 10^6

The amount drained is (7.5×104)(8)=6×105(7.5\times10^4)(8)=6\times10^5 liters. Rewriting this as 0.6×1060.6\times10^6 gives (4.8−0.6)×106=4.2×106(4.8-0.6)\times10^6=4.2\times10^6 liters remaining.

Practice 41 Integer and rational exponents questions in the app

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